Beta Function - Properties

Properties

The beta function is symmetric, meaning that

\Beta(x,y) = \Beta(y,x). \!

When x and y are positive integers, it follows trivially from the definition of the gamma function that:

\Beta(x,y)=\dfrac{(x-1)!\,(y-1)!}{(x+y-1)!} \!

It has many other forms, including:

\Beta(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!
\Beta(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0 \!
\Beta(x,y) = \int_0^\infty\dfrac{t^{x-1}}{(1+t)^{x+y}}\,dt, \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0 \!
\Beta(x,y) = \sum_{n=0}^\infty \dfrac{{n-y \choose n}} {x+n}, \!
\Beta(x,y) = \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1}, \!
\Beta(x,y)\cdot(t \mapsto t_+^{x+y-1}) = (t \to t_+^{x-1}) * (t \to t_+^{y-1}) \qquad x\ge 1, y\ge 1, \!
\Beta(x,y) \cdot \Beta(x+y,1-y) = \dfrac{\pi}{x \sin(\pi y)}, \!


where is a truncated power function and the star denotes convolution. The second identity shows in particular . Some of these identities, e.g. the trigonometric formula, can be applied to deriving the volume of an n-ball in Cartesian coordinates.

Euler's integral for the beta function may be converted into an integral over the Pochhammer contour C as

This Pochhammer contour integral converges for all values of α and β and so gives the analytic continuation of the beta function.

Just as the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices:

Moreover, for integer n, can be integrated to give a closed form, an interpolation function for continuous values of k:

The beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano. It also occurs in the theory of the preferential attachment process, a type of stochastic urn process.

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